In this section a summary of the seven different methods included for comparison in this chapter is presented.
9.2.1 Probability plot
A widely accepted approach to PCI computation is to use a normal probability plot13 so that the normality assumption can be verified simultaneously. Analogous to the normal probability plot, where the natural process width is between the 0.135 and 99.865 percentiles, surrogate PCI values may be obtained via suitable probability plots:
Cp = USL−LSL
upper 0.135% point−lower 0.135% point
= USL−LSL Up−Lp ,
whereUpandLpare respectively the 99.865 and 0.135 percentiles of the observations.
These percentile points can easily be obtained from the simple computer code that performs probability ploting. Since the median is the preferred central value for a skewed distribution, the equivalentCpuandCplare defined as
Cpu = USL− median x0.998 65− median, Cpl = median −LSL
median−x0.998 65.
Cpkis then taken as the minimum of (Cpu,Cpl).
9.2.2 Distribution-free tolerance intervals
Chan et al.6 adopted a distribution-free tolerance interval approach to computeCp
andCpkfor a nonnormal process using Cp = USL−LSL
6σ = USL−LSL
3 2(4σ)
= USL−LSL 3 (2σ) . This results in
Cp = USL−LSL
w = USL−LSL
3 2w2
= USL−LSL 3w3 ,
Cpk = min [(USL−μ),(μ−LSL)]
w/2 ,
wherewis the width of the tolerance interval with 99.73% coverage 95% of the time, w2is the width of the tolerance interval with 95.46% coverage 95% of the time, andw3
is the width of the tolerance interval with 68.26% coverage 95% of the time. The order statistics estimates ofw,w2 andw3, based on the normality assumption, are given by Chan et al.6 This is a more conservative method, since the natural process width is greater as it is estimated taking the sampling variation into account. This is the only method considered here that will give a different result even if the underlying distribution is normal. It is included here to investigate whether its conservative nature is preserved under nonnormality.
9.2.3 Weighted variance method
Choi and Bai8proposed a heuristic weighted variance method to adjust the PCI values according to the degree of skewness of the underlying population. Let Px be the probability that the process variableXis less than or equal to its meanμ,
Px = 1 n
n i=1
I( ¯X−Xi),
whereI(x)=1 ifx>0 andI(x)=0 ifx<0. The PCI based on the weighted variance method is defined as
Cp= USL−LSL 6σWx whereWx=√
1+ |1−2Px|. Also Cpu =USL−μ
2√ 2Pxσ, Cpl = μ−LSL
3
2(1−Px)σ. 9.2.4 Clements’ method
Clements4replaced 6σ in equation (9.1) byUp−Lp: Cp= USL−LSL
Up−Lp ,
whereUpis the 99.865 percentile andLpis the 0.135 percentile.14ForCpk, the process meanμis estimated by the medianM, and the two 3σs are estimated byUp−Mand M−Lprespectively, giving
Cpk =min
USL−M
Up−M ,M−LSL M−Lp .
Clements’ approach uses the classical estimators of skewness and kurtosis that are based on third and fourth moments respectively, which may be somewhat unreliable for very small sample sizes.
9.2.5 Box--Cox power transformation
Box and Cox15proposed a useful family of power transformations on the necessarily positive response variableXgiven by
X(λ)=
⎧⎨
⎩ Xλ−1
λ , forλ=0,
1nX, forλ=0. (9.3)
This continuous family depends on a single parameterλwhich can be estimated by the method of maximum likelihood as follows.
First, a value ofλfrom a selected range is chosen. For the chosenλwe evaluate Lmax= −12ln ˆσ2+lnJ(λ,X)
= −12ln ˆσ2+(λ−1) n
i=1
lnXi,
where J (λ,X)=
n i=1
∂Wi
∂Xi = n i=1
Xλ−i 1, for allλ, so that lnJ(λ,X) = (λ−1)n
i=1lnXi. The estimate of ˆσ2for fixedλis ˆσ2=S(λ)/n, where S(λ) is the residual sum of squares in the analysis of variance ofX(λ). After calculating Lmax(λ) for several values ofλwithin the range, Lmax(λ) can be plotted againstλ. The maximum likelihood estimator ofλis obtained from the value ofλthat maximizesLmax(λ). With the optimalλ* value, each of theXdata specification limits is transformed into a normal variate using equation (9.3). The corresponding PCIs are calculated from the mean and standard deviation of the transformed data using equations (9.1) and (9.2).
9.2.6 Johnson transformation
Johnson16 developed a system of distributions based on the method of moments, similar to the Pearson system. The general form of the transformation is given by
z=γ+ητ(x;ε, λ), η >0,−∞< γ <∞, λ >0,−∞< ε <∞, (9.4) wherezis a standard normal variate andx is the variable to be fitted by a Johnson distribution. The four parameters, γ,η, ε, and λ are to be estimated, and τ is an arbitrary function which may take one of the following three forms.
9.2.6.1 The lognormal system (SL) τ1(x;ε, λ)=log
x−ε λ
, x≥ε (9.5)
This is the Johnson SL distribution that covers the lognormal family. The required estimates for the parameters are
ηˆ=1.645
log
x0.95−x0.5 x0.5−x0.05
−1
, (9.6)
γˆ*=ηˆlog
1−exp (−1.645/η)ˆ x0.5−x0.05
, (9.7)
εˆ=x0.5=exp
−γˆ*/ηˆ
, (9.8)
where the 100αth data percentile is obtained as theα(n+1)th-ranked value fromn observations. If necessary, linear interpolation between consecutive values may be used to determine the required percentile.
9.2.6.2 The unbounded system (SU) τ2(x;ε, λ)=sinh−1
x−ε λ
, −∞<x<∞. (9.9)
Curves in theSUfamily are unbounded. This family covers thetand normal distribu- tions, among others. For the fitting of this distribution, Hahn and Shapiro17gave tables for the determination of ˆγ and ˆηbased on given values of kurtosis and skewness.
9.2.6.3 The bounded system (SB) τ3(x;ε, λ)=log
x−ε λ+ε−x
, ε≤x≤ε+λ. (9.10)
TheSBfamily covers bounded distributions, which include the gamma and beta dis- tributions. Since the distribution can be bounded at either the lower end (ε), the upper end (ε+λ), or both, this leads to the following situations.
r Case I. Range of variation known. For the case where the values of both endpoints are known, the parameters are obtained as
ηˆ= z1−α−zα log
(x1−α−ε)(ε+λ−xα) (xα−ε)(ε+λ−x1−α)
, (9.11)
γˆ =z1−α−ηˆlog
x1−α−ε ε+λ−x1−α
. (9.12)
r Case II. One endpoint known. In this case an additional equation obtained by matching the median of the data is needed to supplement equations (9.11) and (9.12). This equation is given by
λˆ =(x0.5−ε)
(x0.5−ε) (xα−ε)+(x0.5−ε) (x1−α−ε)−2 (xα−ε) (x1−α−ε)
×
(x0.5−ε)2−(xα−ε) (x1−α−ε)−1
.
r Case III. Neither endpoint known. For the case where neither endpoint is known, four data percentiles have to be matched with the corresponding percentiles of the standard normal distribution. The resulting equations fori=1, 2, 3, 4,
zi =γˆ+ηˆlog
xi−εˆ εˆ+λˆ−xi
,
are nonlinear and must be solved by numerical methods.
The algorithm developed by Hill et al.18is used to match the first four moments ofX to the above distribution families. PCIs are calculated using equations (9.1) and (9.2).
9.2.7 Wright’s process capability index Cs
Wright7proposed a CPI,Cs, that takes into account the skewness by incorporating an additional skewness correction factor in the denominator ofCpmk,5and is defined as
Cs= min (USL−μ, μ−LSL) 3
σ2+(μ−T)2+ |μ3/σ| = min (USL−μ, μ−LSL) 3
σ2+ |μ3/σ| , whereT=μandμ3is the third central moment.
Some of the methods described above have been widely applied in industry, such as probability plotting and Clements’ method; however, method such as the Box-- Cox transformation is relatively unknown to practitioners. It should be noted that when the underlying distribution is normal, theoretically, all the above methods, with the exception of the distribution-free method, should give the same result as the conventionalCpandCpkgiven in equations (9.1) and (9.2). Nevertheless, as they use different statistics and/or different ways of estimating the associated statistics, the resultant estimates for Cp and Cpk will exhibit some variability. In particular, the variability ofCp could be reduced with increasing sample size, as its sampling distribution isχn2−1under the normality assumption. However, the variability ofCpk
can be quite significant for all reasonable sample sizes, as it also depends on the variability in the process shift.19